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An integral mean estimate for polynomials. (English) Zbl 0543.30002

The author’s main result is Theorem. If P(z) is a polynomial of degree n having all its zeros in \(| z| \leq 1\), then for each \(q>0\) \[ (1)\quad n(\int^{2\pi}_{0}| P(e^{i\theta})|^ qd\theta)^{1/q}\leq(A_ q)^{1/q}\max_{| z| =1}| P'(z)|, \] where \(A_ q=2^{q+1}\sqrt{\pi}\Gamma({1\over2}q+{1\over2})/\Gamma({1\over2}q+1).\) The result is best possible and equality in (1) holds for \(P(z)=\alpha z^ n+\beta\) where \(| \alpha | =| \beta |.\) For C- polynomials the reviewer [Proc. Am. Math. Soc. 80, 78-82 (1980; Zbl 0441.30010)] has established a general inequality \[ (2)\quad n\| P\|_ r\leq \| P'\|_{rp}\| 1+z\|_{rq}, \] where \(1/p+1/q=1,\quad p>1,\quad q>1,\quad r>0,\quad \| f\|_ k=(\int^{2\pi}_{0}| f(e^{i\theta})|^ kd\theta)^{1/k}.\) Recently the reviewer has generalized (2) to be valid for P(z) having zeros in \(| z| \leq 1\).
Reviewer: Z.Rubinstein

MSC:

30A10 Inequalities in the complex plane
30C10 Polynomials and rational functions of one complex variable

Keywords:

C-polynomials

Citations:

Zbl 0441.30010
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References:

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